The complexity associated with solving a model depends, to a degree, on the solution method employed. A common approach applied to dissipative partial differential equations (PDEs), such as the DFN, SPMe and SPM models, is the so-called method of lines [1]}. This approach requires only the spatial derivatives to be discretised, leading to a large set of coupled time-dependent ordinary differential equations (ODEs) and algebraic equations that can then be solved using a standard package, such as MATLAB's ode15s. Different techniques can be employed to discretise the spatial operators in the governing PDEs, examples being: finite volume methods [2]}, control volume methods [3]}, and finite element methods [4]}, [5]}. Depending on the nature of the PDEs that comprise the model, different systems of temporal equations arise from the spatial discretisation. If all the PDEs have a time derivative, then we obtain a system of ordinary differential equations (ODEs), while if none has time derivatives we obtain a system of algebraic equations. A mixed scenario is also often encountered in which some equations have time derivatives and others do not; after discretisation this yields a system of differential-algebraic equations (DAEs). Numerical solution of DAEs is more complex than either the solution of a system of ODEs, or that of a system of algebraic equations, and this plays a role in the complexity of battery models that we discuss.
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